3.3. TRANSPORT AND SCATTERING 99
is called emission. Thus, both absorption or emission of energy can occur if the perturbation has
a time dependence exp(iωt). If the potential is time independent (defects of various kinds), the
scattering is elastic (Ei=Ef).
The dominant scattering of carriers involves lattice vibrations resulting from thermal energy.
Carriers may scatter from various crystal imperfections including dopants and other point de-
fects, alloy disorder, and interface imperfections.
Phonon scattering
In chapter 1, we discussed the crystalline structure in which atoms were at fixed periodic po-
sitions. In reality, the atoms in the crystal are vibrating around their mean positions. These
lattice vibrations are represented by “particles” in quantum mechanics and are called phonons.
The properties of the lattice vibrations are represented by the relation between the vibration am-
plitude,u, frequency,ω, and the wavevectorq. The vibration of a particular atom,i,isgiven
by
ui(q)=uoiexpi(q·r−ωt) (3.3.11)
which represents an oscillation with quantum energyω. In a semiconductor there are two kinds
of atoms in a basis. This results in a typicalωvs.krelation shown in figure 3.7. Although the
results are for GaAs, they are typical of all compound semiconductors. We notice two kinds of
lattice vibrations, denoted by acoustic and optical. Additionally, there are two transverse and
one longitudinal modes of vibration for each kind of vibration. The acoustic branch can be
characterized by vibrations where the two atoms in the basis of a unit cell vibrate with the same
sign of the amplitude as shown in figure 3.7b. In optical vibrations, the two atoms with opposing
amplitudes are shown.
As noted above, in quantum mechanics lattice vibrations are treated as particles carrying en-
ergyω. According to the discussion on Bose-Einstein statistics in chapter 2, the phonon occu-
pation is given by
nω=1
exp(
ω
kBT)
− 1
(3.3.12)
According to quantum mechanics, the total energy contained in the vibration is given by
Eω=(nω+1
2
)ω (3.3.13)Note that even if there are no phonons in a particular mode, there is a finite “zero point” energy
1
2 ωin the mode. This is important since even ifn=0one can have scattering processes.
The vibrations of the atoms produce three kinds of potential disturbances that result in the
scattering of electrons. A schematic of the potential disturbance created by the vibrating atoms
is shown in figure 3.8. In a simple physical picture, we can imagine the lattice vibrations causing
spatial and temporal fluctuations in the conduction and valence band energies. The electrons
(holes) then scatter from these disturbances. The acoustic phonons produce a strain field in the
crystal and the electrons see a disturbance which produces a potential of the form
VAP=D
∂u
∂x